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In this book, we present some results in the area of the minimization problems of some functionals and boundary value problems with application in the area of optimal control theory concerning elliptic and parabolic partial differential systems with boundary Neumann conditions involving self-adjoint second order operators with infinite number of variables. First, we state and prove some characterization of the minimizing element of some functionals and boundary value problems. As an application of our results, we prove the existence of solution for some elliptic partial differential systems defined on Hilbert spaces with Neumann conditions involving second order operators with infinite number of variables; then we find the set of inequalities that defining the optimal control of these systems.